Subgroups — #36379643

Order 22599.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419230	C₃	1	2	yes	maximal, complemented
2	C₃₁⋊#6855	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃₁×#732	C₃	1	1	yes	maximal, characteristic
4	C₃²×C₃₁⋊C₉	—	3	1	no	—
5	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
6	C₃×H(3,𝔽₃)×C₃₁	—	3	1	no	—
7	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
8	C₃₁×#732	—	3	3	no	—
9	C₃³×C₃₁	—	9	1	no	—
10	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
11	H(3,𝔽₃)×C₃₁	—	3	3	no	—
12	H(3,𝔽₃)×C₃₁	—	9	2	no	—
13	C₃×C₉×C₃₁	—	3	1	no	—
14	C₃×C₉×C₃₁	—	3	2	no	—
15	C₃×C₉×C₃₁	—	3	1	no	—
16	C₃×C₃₁⋊C₉	—	3	3	no	—
17	C₃×C₃₁⋊C₉	—	9	1	no	—
18	C₃×C₃₁⋊C₉	—	9	1	no	—
19	C₃×C₃₁⋊C₉	—	9	1	no	—
20	C₃×C₃₁⋊C₉	—	9	2	no	—
21	#10624489	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	—	3	1	no	—
23	C₃²×C₃₁	—	3	1	no	—
24	C₃²×C₃₁	—	3	1	no	—
25	C₃²×C₃₁	—	9	3	no	—
26	C₃²×C₃₁	—	27	1	no	—
27	C₃²×C₃₁	—	27	1	no	—
28	C₃²×C₃₁	—	27	1	no	—
29	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃²×C₃₁	#731	1	2	yes	—
31	C₃²×C₃₁	#732	1	1	yes	characteristic
32	C₉×C₃₁	—	9	1	no	—
33	C₃₁⋊C₉	—	9	3	no	—
34	C₃₁⋊C₉	—	27	2	no	—
35	C₃×#732	C₃₁⋊C₃	1	1	yes	characteristic
36	#6868	—	31	2	no	—
37	#6855	—	31	1	no	—
38	C₃×C₃₁	—	3	1	no	—
39	C₃×C₃₁	—	3	2	no	—
40	C₃×C₃₁	—	27	3	no	—
41	C₃×C₃₁	#6859	1	1	yes	characteristic
42	C₃×C₃₁	#6879	1	1	yes	characteristic
43	C₃×C₃₁	#6883	1	1	yes	characteristic
44	C₃×C₃₁	#6883	1	1	yes	characteristic
45	C₃×H(3,𝔽₃)	—	3	1	no	—
46	C₃²×C₉	—	93	1	no	—
47	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
48	#732	—	3	3	no	—
49	C₃²⋊C₉	—	93	2	no	—
50	C₃₁	#10624489	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	9	1	no	—
52	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
53	H(3,𝔽₃)	—	3	3	no	—
54	H(3,𝔽₃)	—	9	2	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	3	2	no	—
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	279	1	no	—
59	C₃×C₉	—	279	1	no	—
60	C₃×C₉	—	279	1	no	—
61	C₃×C₉	—	279	2	no	—
62	C₃×C₉	—	93	3	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	9	3	no	—
67	C₃²	—	27	1	no	—
68	C₃²	—	27	1	no	—
69	C₃²	—	27	1	no	—
70	C₃²	#14578894	1	1	yes	characteristic
71	C₃²	#14578896	1	2	yes	—
72	C₃²	#14578899	1	1	yes	characteristic
73	C₉	—	9	1	no	—
74	C₉	—	279	3	no	—
75	C₉	—	837	2	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	2	no	—
78	C₃	—	27	3	no	—
79	C₃	#21419209	1	1	yes	characteristic
80	C₃	#21419264	1	1	yes	characteristic
81	C₃	#21419282	1	1	yes	characteristic
82	C₃	#21419283	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#14578899
commutator, cosocle	C₃²×C₉×C₃₁	C₃²
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting, prime radical	C₃×C₃₁×#732	C₃
socle	C₃²×C₃₁	#732
hypercenter	C₃×#732	C₃₁⋊C₃
hypocenter	C₃₁	#10624489
LCS	G ▷ C₃²×C₉×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#732

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #21419230	C₃	semidirect
2. C₃₁⋊#6855	C₃	semidirect
50. C₃₁	#10624489	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21419230	3
C₃₁⋊#6855	3
C₃×C₃₁×#732	3
#10624489	31